The main remaining policy and process roles of the Monarch are:
(1) As an arbitrator of ambiguities in which parties and individuals should be allowed to become prime minister and appoint a cabinet or to organize an opposition respectively.
For example, if two parties have an equal plurality non-majority number of seats and none of the third parties are willing to join either party in government, the Queen would decide what to do. Similarly the Queen would have the last say in determining if a prime minister is incapacitated or disgraced and must be removed or resign. The Queen also prevents lame duck governments from overstepping their bounds by not recognizing their actions if they do.
(2) To intervene as a definitive voice leading the way in any constitutional crisis that might emerge.
For example, declaring which faction in a multi-modal civil war like the one is Syria is legitimate. Or, declaring which of multiple military factions in a coup is legitimate. Or, determining which reconstruction factions are legitimate if all government collapses in an apocalypse.
Theoretically, the monarch's ratification of ordinary government actions is a structure which also assures the regularity and legitimacy of existing custom and practice in a way not easily achieved when a mere paper document is vested with supreme legal authority as in the U.S. (which could be reinterpreted in a way that renders a long history of past practice retroactively unconstitutional).
The monarch is also generally kept "in the loop" by political leaders, receives briefings, and privately renders advice to existing political leaders (without contradicting them in public in ordinary circumstances).
The monarch's effective role is similar to that of a board of directors in a modern business corporation that formally ratifies a lot, but practically only intervenes in CEO succession and times of crisis and ambiguity.
I will have to leave it to someone else to quote chapter and verse of legislation supporting this, as I've read this mostly in secondary sources and as the drafting style of the legislation in question (some of which I have seen) has a meaning that isn't obvious without the gloss of what is meant by terms like "the Crown" in the law of the United Kingdom.
UPDATED: The Queen also has more or less unfettered discretion regarding the management and disposition of her personal as opposed to sovereign wealth and in matters of taste and style.